Questions tagged [central-limit-theorem]

This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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19 views

Trimmed second moment going to zero by LDCT

We have $0<\sigma<\infty$ and $\epsilon >0$ and $X_1, X_2,...$ iid. The argument involves CLT and it continues on until the line below: That $\frac{1}{\sigma^2}E\big[(X_1-\mu)^2\mathbb{1}\...
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26 views

limiting distribution of Xbar considering an autoregressive process

Here is the question: Consider the stationary Gaussian autoregressive process of order 1, $X_{i+1} − μ = ρ(X_i − μ) + \sigma Z_i$, where $Z_i$ are iid N(0, 1). Find the limiting distribution of $\...
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92 views

A die is thrown until the first time the total sum of the face values of the die is 700 or greater. What is the probability for $n$ tosses?

Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required We're supposed to be using ...
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2answers
64 views

Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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38 views

Limit of a Symmetric Random Walk

I'm given a probability space of ($\Omega$, $\mathcal{F}$, $\mathbb{P}$) and am asked to look into a symmetric random walk with its n-step defined as $$ X_k = \Bigg\{ \begin{matrix} 1 & \text{...
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36 views

central limit theorem: what is the variance?

This is a very basic question that I'm pretty sure I understand but I wanted to double check. Given a regression model of: $$y_t = \mathbf{x_{t}^{\prime}}\beta + u_t$$ We can use one of the CLT ...
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160 views

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables, where $p\in(0,1)$. I am trying to do this by computing the moment generating function of the object I want ...
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1answer
40 views

Generating a number belonging to N(0,1) using *m* numbers from U(0,1) using central limit theorem

I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ. As a starting point, author uses generated uniform ...
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1answer
36 views

Probability dice flip Central Limit Theorem problem

Suppose you flip a fair dice 300 times. Let $X$ be the number of times a $6$ was thrown. What is the probability that $X$ is greater than 60? How can I start with such a problem?
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50 views

Can the Central Limit Theorem be applied here?

My problem statement is to identify in a healthcare organisation, which of it's doctors are lagging in providing proper care to their patients. My Random Variable X is defined as {0 if the patient ...
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1answer
48 views

Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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38 views

Probability of making it on the train if $100$ people stand in line in front of you and each person takes time according to exponential law

The train is leaving in $10$ minutes and there are $100$ people standing in line before you to buy tickets. Each person buys $1.85$ tickets on average with a standard deviation of $1.5$ and $9$ people ...
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37 views

General Central Limit Theorem for Binomial Random Variables

Question Let $(X_n)_{n\geq 1}$ be a sequence of arbitrary binomial random variables such that $EX_n\to \infty$ and $\text{Var}(X_n)/EX_n^2\to 0$ as $n\to \infty$. Then show that $$ Z_n=\frac{...
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168 views

Law of Large Numbers contradicts Central Limit Theorem?

My text defines the weak law of large numbers: If $X_1,\ldots,X_n$ are IID, then $\overline{X} \overset{P}{\to} \mu$. And the CLT as: Let $X_1,\ldots,X_n$ be IID with mean $\mu$ and variance $\...
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89 views

My simulation of the Central Limit Theorem does not converge to correct value

The Lindeberg–Lévy CLT states: Assume $\\{ X_1, X_2, \dots \\}$ is a sequence of i.i.d. random variables with $\mathbb{E}[X_i] = \mu$ and $\text{Var}[X_i] = \sigma^2 < \infty$. And let $S_n = \...
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1answer
67 views

What's wrong with my simulation of the Central Limit Theorem?

While there is code in this question, I suspect the answer will be mathematical. I am trying to create a numerical simulation of the Central Limit Theorem (CLT). My understanding is that if $S_n = \...
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33 views

How does the CLT justify statistical models which are not modeling our data as a sum of random variables?

Wikipedia says (emphasis mine): The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal ...
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30 views

Let $X_1,…, X_n$ be a sample from the normal mixture $(1-p)N(0, 1)+pN(\theta, 1)$

Let $Z_i(\theta)=e^{X_i \theta-\frac{1}{2}\theta^2}-1$. Is $\sum Z_i(\theta)=O_p(\sqrt(n))$, $\sum Z^2_i(\theta)=O_p(1)$ and $\sum Z^3_i(\theta)=O_p(1)$ and why?
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64 views

Proving that CLT doesn't hold for a given sequence.

During examination of compound Poisson process, with log-normal distribution I came across to the following problem. I have examined the following form $$L=\sum_{i=1}^{N}X_{i}$$ And $X_{i}\sim LogN(\...
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1answer
56 views

Show $Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$

Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly ...
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1answer
170 views

If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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27 views

Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
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14 views

An example where Lyapounov Condition is much easier than the Lindeberg?

Currently studying the Lindeberg-Levy-Feller and Lyapounov's Central Limit Theorems, and was wondering - for sake of justification - are there any examples where verifying Lyapounov's condition is ...
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1answer
103 views

A central limit theorem for dependent random variable.

Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e. $$ u_j\stackrel{d}{=}\text{N}(0,1). $$ Call $\mu_r=\mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution ...
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31 views

CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...
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1answer
38 views

Central limit theorem … need help

So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, ...
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30 views

Proof that if you take enough steps of equal magnitude on a plane, you'll always end up at the starting point

Is the claim in the title true? If yes, is the following sufficient to justify it intuitively? Assume for simplicity that the steps are performed with magnitude 1 in a Cartesian plane. To construct a ...
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2answers
59 views

Probability limits of random variable sums

I have $X_1, X_2, X_3, \cdots$ which are independent random variables with the same non-zero mean ($\mu\ne0$) and same variance $\sigma^2$. I would like to compute $$\lim_{n\to\infty} P[\frac{1}n\sum^...
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19 views

Central limit theorem. Calculating probability P(N≤49)

Here is the problem: Apples are being packed in a box. One apple weight is expected to be 200 g with a dispersion of 20 g. Packing is stopped as soon as the total weight is 10 kg or more. Calculate ...
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1answer
24 views

A Central Limit Theorem simple example

A disscusion in the book: Let $(X_n)_{n=1}^\infty$ a sequence of i.i.d random variables such that $\mathbb{E}[X_n]=60, \operatorname{Var}[X_n]=25$. Let $S_N= \sum_{i=1}^NX_i$. By the central limit ...
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57 views

Normal approximation of sum of uniform independent RVs using CLT

Let $X_1$, $X_2$, ... $X_{16}$ and $Y_1$, $Y_2$, ... $Y_{16}$ be independent uniform random variables over the interval [-1,1] and let: $$ W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16} ...
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1answer
68 views

Central Limit Theorem for not identical distributed but independent centered random variables with variance one.

so let's assume we have independent random variables $X_1,X_2, X_3, \ldots$ with $$\mathbb{E}[X_k]=0 \mbox{ and } \mathbb{Var}[X_k]=\sigma_k^2=1 \quad \forall k\in\mathbb{N}. $$ We define $$s_n^2:= ...
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57 views

Does a sequence of random variables constructed in a certain manner converge in distribution to a Gaussian?

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of of IID random variables taken for simplicity with mean zero and variance one. The Central Limit Theorem give us that $$ \frac{X_1 + \dots + X_n}{\...
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1answer
39 views

using Central Limit Theorem to approximate a probability with large 'n'

For $i>1$ , let $X_i$ ~ $G_{1/2}$ be distributed Geometrically with parameter $1/2$. $$ Y_n= \frac{1}{\sqrt n} \sum_{r=1}^n X_r-2 $$ Approximate $P(-1\le Y_n\le 2)$ with large n.($Y_n$ is not ...
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92 views

Let $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta\in(0,1)$, then $S_n/n^{(3-\beta)/2)}\Rightarrow c\chi$

Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$, where $\beta>0$. Show that: (i) If $\beta>1$ then $S_n\to S_\infty$ a.s. (ii) If $\beta\in(0,1)$ then $S_n/n^{(3-...
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1answer
40 views

Central Limit Theorem using sample standard deviation

Let $X_1, X_2,..$ be iid random variables with mean $\mu$ and variance $\sigma^2$. Show $$ \displaystyle \frac{\sum_{i = 1}^n (X_i - \mu)}{\sqrt{\sum_{i = 1}^n (X_i - \overline{X}_n)^2}} \to N(0,1) $...
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29 views

Question about limiting distribution for standard normal distribution

I am studying CLT and meet a question but I have no idea how to solve it, could you please show me how to prove this question (b)? Thank you so much! Wang
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166 views

Central limit theorem for sequence of Gamma-distributed random variables.

Suppose that $X_ n \sim \text {Gamma}\ (n\alpha , \lambda)$ for all $n \ge 1$, for fixed $\alpha,\lambda >0.$ Show that $$\frac {1} {\sqrt n} \left (X_n - \frac {n \alpha} {\lambda} \right ) \...
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1answer
27 views

Central Limit Theorem and convergence of transformed

I have the following exercise to solve: Let $X_n, n \geq1$ be a sequence of i.i.d random variables where each $X_n$ is a discrete random variable with distribution $P(X_n=1)=1-p$ and $P(X_n=2)=p$, ...
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1answer
42 views

Central Limit Theorem - Different Forms

Given the following function: $W_n = \frac{1}{\sqrt{n}}\Pi_{k=1}^{\infty}\log(U_k)$ where $U_k$ is uniformly distributed from $1$ to $e$. Does $\{W_n\}_{n\geq 1}$ converge in distribution? I found ...
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88 views

approximate probability of geometric distribution using CLT

I have the following problem: For $i≥1$, let $X_i∼G_1/2$ be distributed Geometrically with parameter 1/2. Define $$Y_n=\frac{1}{\sqrt{n}}\sum_{i=1}^n (X_i-2)$$ Approximate $P(−1≤Y_n≤2)$ with large ...
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10 views

$X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large numbers and the central limit theorem are valid

$Z_1,Z_2,...$ are i.i.d., their expected value is zero, their variance $\sigma^2$, and $E[|Z_i^2|] = m_3 < \infty$. $X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large ...
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1answer
16 views

Confidence Interval of number of red marbles among 100 marbles where proportion of red marbles is uniformly distributed

A bag contains 100 marbles of colors red and black. The proportion of red color marbles is uniformly distributed between 0 and 1. How do I compute the confidence interval of the number of red marbles? ...
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2answers
30 views

What is the limit of the following expression?

I've been thinking about and trying to solve the following limit that I just feel lost by now. I always get an indeterminate form. I don't know what else to try. In the picture is just one way that I ...
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1answer
153 views

Proof of Convergence in Distribution for random variables with infinite variance

We are asked to prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true: $$ \frac{X_1+X_2 + \dots +X_n}{\sqrt{n\log n}} \xrightarrow{\...
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1answer
33 views

Application of Central Limit Theorem to Sales

Consider the following problem and solution. (I am stuck at the modified problem.) Problem There are exactly two phone shops, $A$ and $B$, serving a town of 1000 people. Both shops sell an iPhoneX ...
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27 views

Approximating a random variable versus approximating probability statements about a random variable?

After formally stating the central limit theorem my statistics textbook says this: Interpretation: Probability statements about the sample mean $\overline{X}_n$ can be approximated using a Normal ...
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50 views

Bochner's theorem

I'm reading Bochner's theorem. Now I'm having problem with part on the third page: $\int_{s\in [0,T], s+u\in [0, T]} ds=1-\frac{|u|}{T}$? How to deduce it? Any help is welcome. Thanks in advance.
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2answers
147 views

Understanding the central limit theorem

I am an aspiring probabilist, and I definitely know the central limit theorem. However, I am trying to understand what idea it really embodies. I am aware that normal distribution arises as the limit ...
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25 views

Potentially new method for obtaining asymptotic distribution of M-estimators

Disclaimer. I'm not quite sure this is the best venue for this question, but I'll give it a try... So, in a comment to this MO post, it was said that one can use the comment right after Remark 7.3 in ...